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Rigid rotorFrom Wikipedia, the free encyclopedia

The rigid rotor is a mechanical model that is used to explain rotating systems. An arbitrary rigid rotor is a 3-dimensionalrigid object, such as a top. To orient such an object in space three angles known as Euler angles are required. A specialrigid rotor is the linear rotor which requires only two angles to describe its orientation. An example of a linear rotor is adiatomic molecule. More general molecules like water (asymmetric rotor), ammonia (symmetric rotor), or methane(spherical rotor) are 3-dimensional, see classification of molecules.

Contents

1 Linear rotor1.1 Classical linear rigid rotor1.2 Quantum mechanical linear rigid rotor1.3 Selection rules1.4 Non-rigid linear rotor

2 Arbitrarily shaped rigid rotor2.1 Coordinates of the rigid rotor2.2 Classical kinetic energy

2.2.1 Angular velocity form2.2.2 Lagrange form2.2.3 Angular momentum form2.2.4 Hamilton form

2.3 Quantum mechanical rigid rotor3 Direct experimental observation of molecular rotations4 See also5 References6 General references

Linear rotor

The linear rigid rotor model consists of two point masses located at fixed distances from their center of mass. The fixeddistance between the two masses and the values of the masses are the only characteristics of the rigid model. However,for many actual diatomics this model is too restrictive since distances are usually not completely fixed. Corrections on therigid model can be made to compensate for small variations in the distance. Even in such a case the rigid rotor model is auseful point of departure (zeroth-order model).

Classical linear rigid rotor

The classical linear rotor consists of two point masses ) each at a

distance is independent of time. The kinematics of a linear rigid rotor is usually described by

means of . In the physics convention the coordinatesare the co-latitude (zenith) angle orientation of the rotor in space. The kinetic energy

where

Scale factors are of importance for quantum mechanical applications since they enter the curvilinear coordinates

The classical Hamiltonian function of the linear rigid rotor is

Quantum mechanical linear rigid rotor

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Quantum mechanical linear rigid rotor

The linear rigid rotor model can be used in quantum mechanics to predict the rotational energy of a diatomic molecule.The rotational energy depends on the moment of inertia for the system, . In the center of mass reference frame, themoment of inertia is equal to:

where

According to Schrödinger equation:

where ) operator. For the rigid rotor in a field-free space, the

energy operator corresponds to the

where . The Laplacian is given above in terms of spherical polar

coordinates. The energy operator written in terms of these coordinates is:

This operator appears also in the Schrödinger equation of the hydrogen atom after the radial part is separated off. Theeigenvalue equation becomes

The symbol . Note that the energy does not

depend on

is have the same energy.

Introducing the

In the units of

with , a unit that is often

used for rotational-vibrational spectroscopy. The rotational constant . Often one

writes (the value for which the interaction energy of the atoms in

the rotor has a minimum).

A typical rotational spectrum consists of a series of peaks that correspond to transitions between levels with differentvalues of the angular momentum appear at energies corresponding

to an integer multiple of .

Selection rules

Rotational transitions of a molecule occur when the molecule absorbs a photon [a particle of a quantized electromagnetic(em) field]. Depending on the energy of the photon (i.e., the wavelength of the em field) this transition may be seen as asideband of a vibrational and/or electronic transition. Pure rotational transitions, in which the vibronic (= vibrational pluselectronic) wave function does not change, occur in the microwave region of the electromagnetic spectrum.

Typically, rotational transitions can only be observed when the angular momentum quantum number changes by 1 (

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Typically, rotational transitions can only be observed when the angular momentum quantum number changes by 1 (). This selection rule arises from a first-order perturbation theory approximation of the time-dependent

A transition occurs if this integral is non-zero. By separating the rotational part of the molecular wavefunction from the. After integration

will result in nonzero values for the dipole transition moment integral. This

) arenot completely fixed; the bond between the atoms stretches out as the molecule rotates faster (higher values of the

). This frequency is related to the reduced mass

The non-rigid rotor is an acceptably accurate model for diatomic molecules but is still somewhat imperfect. This isbecause, although the model does account for bond stretching due to rotation, it ignores any bond stretching due tovibrational energy in the bond (anharmonicity in the potential).

Arbitrarily shaped rigid rotor

An arbitrarily shaped rigid rotor is a rigid body of arbitrary shape with its center of mass fixed (or in uniform rectilinear

motion) in field-free space R3, so that its energy consists only of rotational kinetic energy (and possibly constanttranslational energy that can be ignored). A rigid body can be (partially) characterized by the three eigenvalues of itsmoment of inertia tensor, which are real nonnegative values known as principal moments of inertia. In microwavespectroscopy—the spectroscopy based on rotational transitions—one usually classifies molecules (seen as rigid rotors)as follows:

spherical rotorssymmetric rotors

oblate symmetric rotorsprolate symmetric rotors

asymmetric rotors

This classification depends on the relative magnitudes of the principal moments of inertia.

Coordinates of the rigid rotor

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Coordinates of the rigid rotor

Different branches of physics and engineering use different coordinates for the description of the kinematics of a rigidrotor. In molecular physics Euler angles are used almost exclusively. In quantum mechanical applications it isadvantageous to use Euler angles in a convention that is a simple extension of the physical convention of spherical polarcoordinates.

The first step is the attachment of a right-handed orthonormal frame (3-dimensional system of orthogonal axes) to therotor (a body-fixed frame) . This frame can be attached arbitrarily to the body, but often one uses the principal axesframe—the normalized eigenvectors of the inertia tensor, which always can be chosen orthonormal, since the tensor isHermitian. When the rotor possesses a symmetry-axis, it usually coincides with one of the principal axes. It is convenientto choose as body-fixed z-axis the highest-order symmetry axis.

One starts by aligning the body-fixed frame with a space-fixed frame (laboratory axes), so that the body-fixed x, y, and zaxes coincide with the space-fixed X, Y, and Z axis. Secondly, the body and its frame are rotated actively over a positive

angle -axis. Thirdly, one rotates the body and

its frame over a positive angle -axis of the body-fixed frame has after these two rotations the

longitudinal angle ), both with

respect to the space-fixed frame. If the rotor were cylindrical symmetric around its -axis, like the linear rigid rotor, itsorientation in space would be unambiguously specified at this point.

If the body lacks cylinder (axial) symmetry, a last rotation around its ) is

necessary to specify its orientation completely. Traditionally the last rotation angle is called

The convention; it can be shown (in the same

manner as in convention in which the order of rotations is reversed.

The total matrix of the three consecutive rotations is the product

Let in the body with respect to the body-fixed frame. The elements

of . Upon rotation

of the body, the body-fixed coordinates of becomes,

In particular, if

which shows the correspondence with the

Knowledge of the Euler angles as function of time determine the kinematics of the rigid

rotor.

Classical kinetic energy

The following text forms a generalization of the well-known special case of the of an object that rotates around one

axis.

It will be assumed from here on that the body-fixed frame is a principal axes frame; it diagonalizes the instantaneous

inertia tensor

where the Euler angles are time-dependent and in fact determine the time dependence of by the inverse of this

equation. This notation implies that at the Euler angles are zero, so that at the body-fixed frame coincideswith the space-fixed frame.

The classical kinetic energy T of the rigid rotor can be expressed in different ways:

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The classical kinetic energy T of the rigid rotor can be expressed in different ways:

as a function of angular velocityin Lagrangian formas a function of angular momentumin Hamiltonian form.

Since each of these forms has its use and can be found in textbooks we will present all of them.

Angular velocity form

As a function of angular velocity T reads,

with

The vector

the body-fixed frame. It can be shown that definition of

velocityequations of motion known as space).

Lagrange form

Backsubstitution of the expression of derivatives of the Euler angles). In matrix-vector notation,

where

Angular momentum form

Often the kinetic energy is written as a function of the

(time-independent) quantity. With respect to the body-fixed frame it has the components related to the angular velocity,

Since the body-fixed frame moves (depends on time) these components are

represent

components. The kinetic energy is given by

Hamilton form

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Hamilton form

The Hamilton form of the kinetic energy is written in terms of generalized momenta

where it is used that the

with the inverse metric tensor given by

This inverse tensor is needed to obtain the

mechanical energy operator of the rigid rotor.

The classical Hamiltonian given above can be rewritten to the following expression, which is needed in the phase integralarising in the classical statistical mechanics of rigid rotors,

Quantum mechanical rigid rotor

See also:

As usual quantization is performed by the replacement of the generalized momenta by operators that give first derivativeswith respect to its

and similarly for of all three Euler

angles, time derivatives of Euler angles, and inertia moments (characterizing the rigid rotor) by a simple differentialoperator that does not depend on time or inertia moments and differentiates to one Euler angle only.

The quantization rule is sufficient to obtain the operators that correspond with the classical angular momenta. There aretwo kinds: space-fixed and body-fixed angular momentum operators. Both are vector operators, i.e., both have threecomponents that transform as vector components among themselves upon rotation of the space-fixed and the body-fixedframe, respectively. The explicit form of the rigid rotor angular momentum operators is given (but beware, they must

be multiplied with

commutation relations.

The quantization rule is not sufficient to obtain the kinetic energy operator from the classical Hamiltonian. Since

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The quantization rule is not sufficient to obtain the kinetic energy operator from the classical Hamiltonian. Sinceclassically and the inverses of these functions, the position of these trigonometric

functions in the classical Hamiltonian is arbitrary. After quantization the commutation does no longer hold and the order

of operators and functions in the Hamiltonian (energy operator) becomes a point of concern. Podolsky

1928 that the

energy operator. This operator has the general form (summation convention: sum over repeated indices—in this case

over the three Euler angles

where

Given the inverse of the metric tensor above, the explicit form of the kinetic energy operator in terms of Euler anglesfollows by simple substitution. (Note: The corresponding eigenvalue equation gives the for the rigid

rotor in the form that it was solved for the first time by Kronig and RabiThis is one of the few cases where the Schrödinger equation can be solved analytically. All these cases were solved withina year of the formulation of the Schrödinger equation.)

Nowadays it is common to proceed as follows. It can be shown that can be expressed in body-fixed angular momentum

operators (in this proof one must carefully commute differential operators with trigonometric functions). The result hasthe same appearance as the classical formula expressed in body-fixed coordinates,

The action of the

so that the Schrödinger equation for the spherical rotor (

degenerate energy equal to

The symmetric top (= symmetric rotor) is characterized by .

In the latter case we write the Hamiltonian as

and use that

Hence

The eigenvalue have the same

eigenvalue. The energies with |k| > 0 are -fold degenerate. This exact solution of the Schrödinger equation of

the symmetric top was first found in 1927.

The asymmetric top problem (

Direct experimental observation of molecular rotations

For long time, molecular rotations could not be directly observed experimentally. Only measurement techniques with

atomic resolution made it possible to detect the rotation of a single molecule.[4][5] At low temperatures, the rotations ofmolecules (or part thereof) can be frozen. This could be directly visualized by Scanning tunneling microscopy i.e., the

stabilization could be explained at higher temperatures the rotational entropy. [5]

See also

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See also

Balancing machineGyroscopeInfrared spectroscopyRigid bodyRotational spectroscopySpectroscopyVibrational spectroscopyQuantum rotor model

References

. 1 ^ a b Podolsky, B. (1928). Phys. Rev. 32: 812. Bibcode:1928PhRv...32...812J(http://adsabs.harvard.edu/abs/1928PhRv...32...812J). doi:10.1103/PhysRev.32.812(http://dx.doi.org/10.1103%2FPhysRev.32.812).

. 2 ^ Chapter 4.9 of Goldstein, H.; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (Third ed.). San Francisco: Addison WesleyPublishing Company. ISBN 0-201-65702-3.

. 3 ^ a b R. de L. Kronig and I. I. Rabi (1927). "The Symmetrical Top in the Undulatory Mechanics". Phys. Rev. 29: 262–269.Bibcode:1927PhRv...29..262K (http://adsabs.harvard.edu/abs/1927PhRv...29..262K). doi:10.1103/PhysRev.29.262(http://dx.doi.org/10.1103%2FPhysRev.29.262).

. 4 ^ J. K. Gimzewski, C. Joachim, R. R. Schlittler, V. Langlais, H. Tang, I. Johannsen (1998), "Rotation of a Single Molecule Within aSupramolecular Bearing" (in German), Science 281 (5376): pp. 531–533, doi:10.1126/science.281.5376.531(http://dx.doi.org/10.1126%2Fscience.281.5376.531)

. 5 ^ a b Thomas Waldmann, Jens Klein, Harry E. Hoster, R. Jürgen Behm (2012), "Stabilization of Large Adsorbates by RotationalEntropy: A Time-Resolved Variable-Temperature STM Study" (in German), ChemPhysChem: pp. n/a–n/a,doi:10.1002/cphc.201200531 (http://dx.doi.org/10.1002%2Fcphc.201200531)

General references

D. M. Dennison (1931). "The Infrared Spectra of Polyatomic Molecules Part I". Rev. Mod. Physics 3: 280–345.Bibcode:1931RvMP....3..280 D (http://adsabs.harvard.edu/abs/1931RvMP....3..280 D). doi:10.1103/RevModPhys.3.280(http://dx.doi.org/10.1103%2FRevModPhys.3.280). (Especially Section 2: The Rotation of Polyatomic Molecules).Van Vleck, J. H. (1951). "The Coupling of Angular Momentum Vectors in Molecules". Rev. Mod. Physics 23: 213–227.Bibcode:1951RvMP...23..213 V (http://adsabs.harvard.edu/abs/1951RvMP...23..213 V).doi:10.1103/RevModPhys.23.213 (http://dx.doi.org/10.1103%2FRevModPhys.23.213).McQuarrie, Donald A (1983). Quantum Chemistry. Mill Valley, Calif.: University Science Books. ISBN 0-935702-13-X.Goldstein, H.; Poole, C. P.; Safko, J. L. (2001). Classical Mechanics (Third ed.). San Francisco: Addison WesleyPublishing Company. ISBN 0-201-65702-3. (Chapters 4 and 5)Arnold, V. I. (1989). Mathematical Methods of Classical Mechanics. Springer-Verlag. ISBN 0-387-96890-3. (Chapter6).Kroto, H. W. (1992). Molecular Rotation Spectra. New York: Dover.Gordy, W.; Cook, R. L. (1984). Microwave Molecular Spectra (Third ed.). New York: Wiley. ISBN 0-471-08681-9.Papoušek, D.; Aliev, M. T. (1982). Molecular Vibrational-Rotational Spectra. Amsterdam: Elsevier.ISBN 0-444-99737-7.

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